Common fixed points of monotone Lipschitzian semigroups in Banach spaces

In this paper, we investigate the existence of common fixed points of monotone Lipschitzian semigroup in Banach spaces under the natural condition that the images under the action of the semigroup at certain point are comparable to the point. In particular, we prove that if one map in the semigroup is a monotone contraction mapping, then such common fixed point exists. In the case of monotone nonexpansive semigroup we prove the existence of common fixed points if the Banach space is uniformly convex in every direction. This assumption is weaker than uniform convexity.